DOCSLIB.ORG

Graphs on Surfaces with Positive Forman Curvature Or Corner Curvature

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graphs on Surfaces with Positive Forman Curvature Or Corner Curvature GRAPHS ON SURFACES WITH POSITIVE FORMAN CURVATURE OR CORNER CURVATURE YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG Abstract. On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with positive corner curvature. Contents 1. Introduction 1 2. Preliminaries 5 2.1. Embedding 9 3. Graphs on surfaces with positive Forman curvature 10 4. Graphs on surfaces with positive corner curvature 13 References 20 1. Introduction The Gaussian curvature of a smooth surface is well studied in differential ge- ometry, which describes the convexity of the surface. For a polyhedron in R3; the discrete Gaussian curvature, as a measure, concentrates on the set of vertices. In the graph theory, the combinatorial curvature of a planar graph, which serves as the discrete Gaussian curvature of a canonical piecewise flat surface, was introduced by [31, 36, 17, 23] respectively. It has been extensively studied in the literature; see e.g. [38, 42, 20, 3, 18, 28, 21, 37, 34, 4, 13, 11, 40, 10, 25, 27, 26, 32, 16]. Let S be a (possibly noncompact) connected surface without boundary. Let (V; E) be a (possibly infinite) locally finite, undirected, simple graph with the set arXiv:2002.03550v1 [math.CO] 10 Feb 2020 of vertices V and the set of edges E: We may regard (V; E) as a 1-dimensional topological space. Let ' :(V; E) ! S be an topological embedding. We denote by F the set of faces induced by the embedding ', i.e. connected components of the complement of the embedding image of (V; E) in S: We write G = (V; E; F ) for the cellular complex structure induced by the embedding, which is called a graph on a surface [30] (or a semiplanar graph [22]). For any σ 2 F; we denote by σ the closure of σ in S; which is called the closed face. We say that a semiplanar graph G = (V; E; F ) is a tessellation of S if the following hold, see e.g. [26]: (i) For any compact set K ⊂ S; K can be covered by finitely many closed faces. (ii) Every closed face is homeomorphic to a closed disk whose boundary consists of finitely many edges of the graph. 1 2 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG (iii) Every edge is contained in exactly two different closed faces. (iv) If two closed faces intersect, then the intersection is either a vertex or the closure of an edge. One can show that a tessellation of S is finite if and only if S is compact. It is called a planar tessellation (resp. a tessellation in the real projective plane) if S is the sphere S2 or the plane R2 (resp. RP 2). In this paper, we always consider tessellations, and call planar tessellations planar graphs for simplicity. For a graph on a surface G = (V; E; F ); two vertices x; y are called neighbors if there is an edge connecting x and y; denoted by x ∼ y: Two elements in V; E; F are called incident if the closures of their embedding images intersect. If one is contained in the closure of the other, then we write like x ≺ e; e ≺ σ; x ≺ σ; for x 2 V; e 2 E; σ 2 F: We denote by jxj (resp. jσj) the degree of a vertex x (resp. a face σ), i.e. the number of neighbors of x (resp. the number of edges incident to σ). In this paper, we only consider semiplanar graphs satisfying the following: for any vertex x and face σ; jxj ≥ 3; jσj ≥ 3: Given a graph G = (V; E; F ) on a surface S; the combinatorial curvature at a vertex x is defined as jxj X 1 Φ(x) = 1 − + : 2 jσj σ2F :x≺σ We endow S with a canonical piecewise flat metric as follows: assign each edge length one, replace each face by a regular Euclidean polygon of side-length one with same facial degree, and glue these polygons along the common edges; see [8] for gluing metrics. The ambient space S equipped with the gluing metric is called the (regular Euclidean) polyhedral surface of G; denoted by S(G): For the metric surface S(G), the generalized Gaussian curvature is a measure concentrated on vertices, whose mass at each vertex x is given by the angle defect K(x); i.e. 2π minus the total angle at x: One easily sees that 1 Φ(x) = K(x); 8x 2 V; 2π where K(x) is the angle defect, i.e. the discrete Gaussian curvature, at the vertex x: If S is compact, the discrete Gauss-Bonnet theorem reads as X (1) Φ(x) = χ(S); x2V where χ(·) is the Euler characteristic of S: A planar graph G has nonnegative combinatorial curvature if and only if the polyhedral surface S(G) is a generalized convex surface in the sense of Alexandrov, see [9, 8, 22]. Higuchi [20] conjectured a discrete Bonnet-Myers theorem that a graph on a surface G with positive combinatorial curvature everywhere is finite. It was confirmed by DeVos and Mohar [13], see e.g. [11, 10, 32, 33, 16] for further developments. Theorem 1.1 ([13]). For a graph G = (V; E; F ) on a surface S, if Φ(x) > 0 for any x 2 V; then G is finite. Moreover, S = S2 or RP 2: For a planar graph G with an embedding, one can define the dual graph G∗: the vertices of G∗ are corresponding to faces of G; the faces of G∗ are corresponding to vertices of G; and two vertices in G∗ are adjacent if and only if the corresponding GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 3 faces in G share a common edge. One can show that for a finite graph G; it is a tes- sellation if and only if so is G∗; see e.g. [4]. For a graph on a surface G with positive combinatorial curvature, the dual graph may not have nonnegative combinatorial curvature; 7-gonal prism (resp. 5-gonal antiprism (Figure 5 (right))) has positive combinatorial curvature everywhere, but the dual, the 7-gonal bipyramid (resp. 5- gonal pseudo-double wheel (Figure 8 (upper right))), has negative curvature at the two apexes. This means that the dual operation doesn't preserve the class of graphs on surfaces with positive combinatorial curvature. We will study some curvature notions of graphs on surfaces, for which the dual operation is closed on the class of graphs with positive curvature. Motivated by Bochner techniques for differential forms in Riemannian geometry, Forman [15] introduced the curvature on general CW complexes, which is now called the Forman curvature. For a compact Riemannian manifold M; let ∆p be the Hodge Laplacian on p-forms, p 2 N0: The Bochner-Weitzenb¨ock formula reads as ∗ ∆p = (rp) rp + Fp; where rp is the Levi-Civita covariant derivative operator on p-forms and Fp de- notes the curvature operator on p-forms. On a CW complex, Forman derived an analogous formula for the discrete Hodge Laplacian and defined the remainder term Fp as the discrete curvature on p-cells. In this paper, we only consider the Forman curvature F1 on 1-cells, i.e. edges, with weight one everywhere. This curvature F1 corresponds to the discrete analog of Ricci curvature, and we will denote it by RicF in this paper. For a graph on a surface G = (V; E; F ); two edges e1 and e2 are called parallel neighbors if one and only one of the following holds: (1) There exists x 2 V such that x ≺ e1; x ≺ e2: (2) There exists σ 2 F such that e1 ≺ σ; e2 ≺ σ: The Forman curvature of an edge e is defined as, see [15], (2) RicF (e) = #fσ 2 F : e ≺ σg+#fx 2 V : x ≺ eg−#fparallel neighbors of eg: We are interested in graphs on surfaces with positive Forman curvature (every- where). Note that a graph on surface with positive Forman curvature may not be a tessellation, see e.g. Figure 1. There are infinitely many nontessellation, pla- 2 Figure 1. The graph C4 embedded in S . Each edge e has RicF (e) = 3: nar graphs with positive Forman curvature (everywhere). For example, the star graph K1;n (n ≥ 1) has positive Forman curvature 3 everywhere. In this paper, we only study the class of tessellations on surfaces with positive Forman curvature, denoted by FC+: One easily checks that the class FC+ is closed under the dual operation. Forman proved discrete analogs of Bochner's theorem and the Bonnet- Myers theorem for \quasi-convex" regular CW complexes, see [15, Theorem 2.8 and Theorem 6.3]. In our setting, for a graph G = (V; E; F ) on a surface S; they state 4 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG 1 that if RicF (e) > 0 for each edge e; then H (S; R) = 0 and the diameter of X is finite, respectively. The above theorems yield the following result. Theorem 1.2 ([15]). Let G = (V; E; F ) be a tessellation on a surface S satisfying 2 2 RicF (e) > 0 for all e 2 E: Then G is finite and S = S or RP : f1 e x2 x1 f2 Figure 2 One of main difficulties for the Bonnet-Myers theorem above is that the total Forman curvature of an infinite graph on a surface is possibly infinite.
Load more

Recommended publications
  • Uniform Panoploid Tetracombs
    Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs.
    [Show full text]
  • Order in Space
    Order in space A.K. van der Vegt VSSD Other textbooks on mathematics, published by VSSD Mathematical Systems Theory, G. Olsder and J.W. van der Woude, x + 210 pp. See http://www.vssd.nl/hlf/a003.htm Numerical Methods in Scientific Computing, J. van Kan, A. Segal and F. Vermolen, xii + 279 pp. See http://www.vssd.nl/hlf/a002.htm By the author of Order in space: From Plastics to Polymers, A.K. van der Vegt, 268 pp. See http://www.vssd.nl/hlf/m028.htm © VSSD Edition on the internet, 2001 Printed edition 2006 Published by: VSSD Leeghwaterstraat 42, 2628 CA Delft, The Netherlands tel. +31 15 27 82124, telefax +31 15 27 87585, e-mail: [email protected] internet: http://www.vssd.nl/hlf URL about this book: http://www.vssd.nl/hlf/a017.htm A collection of digital pictures and/or an electronic version can be made available for lecturers who adopt this book. Please send a request by e-mail to [email protected] All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher. NUR 918 ISBN-10: 90-71301-61-3 ISBN-13: 978-90-71301-61-2 Keywords: geometry 3 PREFACE This book deals with a very old subject. Many centuries ago some people were already fascinated by polyhedra, and they spent much time in investigating regular spatial structures. In the terminology of polyhedra we, therefore, meet the names of Archimedes, Pythagoras, Plato, Kepler etc.
    [Show full text]
  • Assembly of One-Patch Colloids Into Clusters Via Emulsion Droplet Evaporation
    materials Article Assembly of One-Patch Colloids into Clusters via Emulsion Droplet Evaporation Hai Pham Van 1,2, Andrea Fortini 3 and Matthias Schmidt 1,* 1 Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, Universitätsstraße 30, D-95440 Bayreuth, Germany 2 Department of Physics, Hanoi National University of Education, 136 Xuanthuy, Hanoi 100000, Vietnam; [email protected] 3 Department of Physics, University of Surrey, Guildford GU2 7XH, UK; [email protected] * Correspondence: [email protected]; Tel.: +49-921-55-3313 Academic Editors: Andrei V. Petukhov and Gert Jan Vroege Received: 23 February 2017; Accepted: 27 March 2017; Published: 29 March 2017 Abstract: We study the cluster structures of one-patch colloidal particles generated by droplet evaporation using Monte Carlo simulations. The addition of anisotropic patch–patch interaction between the colloids produces different cluster configurations. We find a well-defined category of sphere packing structures that minimize the second moment of mass distribution when the attractive surface coverage of the colloids c is larger than 0.3. For c < 0.3, the uniqueness of the packing structures is lost, and several different isomers are found. A further decrease of c below 0.2 leads to formation of many isomeric structures with less dense packings. Our results could provide an explanation of the occurrence of uncommon cluster configurations in the literature observed experimentally through evaporation-driven assembly. Keywords: one-patch colloid; Janus colloid; cluster; emulsion-assisted assembly; Pickering effect 1. Introduction Packings of colloidal particles in regular structures are of great interest in colloid science. One particular class of such packings is formed by colloidal clusters which can be regarded as colloidal analogues to small molecules, i.e., “colloidal molecules” [1,2].
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
    ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron.
    [Show full text]
  • Regulating the Distortion Degree of the Square Antiprism Coordination
    Electronic Supplementary Material (ESI) for CrystEngComm. This journal is © The Royal Society of Chemistry 2021 Electronic Supplementary Information (ESI) for CrystEngComm Regulating the Distortion Degree of the Square Antiprism Coordination Geometry in the Dy-Na Single Ion Magnets Hui-Sheng Wang,*a Ke Zhang,a Jia Wang,b Zhaobo Hu,b You Song,*b Zaichao Zhang,c and Zhi-Quan Pana aSchool of Chemistry and Environmental Engineering, Key Laboratory of Green Chemical Process of Ministry of Education, Key Laboratory of Novel Reactor and Green Chemical Technology of Hubei Province, Wuhan Institute of Technology, Wuhan 430205, P. R. China. bState Key Laboratory of Coordinate Chemistry, Nanjing National Laboratory of Microstructures, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, P. R. China. cJiangsu Key Laboratory for the Chemistry of Low-dimensional Materials, School of Chemistry and Chemical Engineering, Huaiyin Normal University, P. R. China. *Email: [email protected] (H.-S. W.). *Email: [email protected] (Y. S.) -S1- Contents Table S1. Crystal data and structure refinement parameters for complexes 1-2. Table S2. Selected bond lengths (Å) and angles (°) for 1. Table S3. Selected bond lengths (Å) and angles (°) for 2. Table S4. The possible geometries of octa-coordination metal centers and Deviation parameters from each ideal polyhedron for Dy in complex 1. Table S5. The possible geometries of octa-coordination metal centers and Deviation parameters from each ideal polyhedron for Dy in complex 2. Table S6. The details for parameters involved in SAP geometry for 1 and 2. Table S7. Best fitted parameters obtained for the extended Debye model with ac susceptibility data from SQUID magnetometer of compound 1 under a 1000 Oe dc applied field.
    [Show full text]
  • This Is a Set of Activities Using Both Isosceles and Equilateral Triangles
    This is a set of activities using both Isosceles and Equilateral Triangles. 3D tasks Technical information This pack includes 25 Equilateral Triangles and 25 Isosceles Triangles. You will also need a small tube of Copydex. If you are new to making models with Copydex we suggest you start by watching www.atm.org.uk/Using-ATM-MATs Different polyhedra to make Kit A: 4 Equilateral Triangle MATs. Place one triangle on your desk and join the other three triangles to its three sides, symmetrically. Now add glue to one edge of each of the three added triangles so that all can be joined into a polyhedron This is your first polyhedron. Kit B: 3 Isosceles triangles and 1 equilateral triangle Kit C: 4 Isosceles triangles Kit D: 6 Equilateral Triangles Kit E: 6 Isosceles Triangles Kit F: 3 Equilateral and 3 Isosceles Triangles. (are there different possible polyhedra this time?) Kit G: 8 Equilateral Triangles. One possibility is a Regular Octahedron, but is there more than one Octahedron? Kit H: 4 Equilateral and 4 Isosceles Triangles (are there different possible polyhedra this time?) Kit I: 2 Equilateral Triangles and 6 Isosceles Triangles Kit J: Go back to the Regular Octahedron (from Kit G) and add a Regular Tetrahedron to one of its faces, then add another … until four have been added. Keep track (in a table) of how many faces and vertices the Regular Octahedron and the next four polyhedra have. Hint: if you spaced the added Tetrahedra equally around the Octahedron you will find a bigger version of a previously made solid.
    [Show full text]
  • Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected]
    Bridgewater State University Virtual Commons - Bridgewater State University Mathematics Faculty Publications Mathematics Department 2015 Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected] Virtual Commons Citation Burgiel, Heidi (2015). Unit Origami: Star-Building on Deltahedra. In Mathematics Faculty Publications. Paper 47. Available at: http://vc.bridgew.edu/math_fac/47 This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture Unit Origami: Star-Building on Deltahedra Heidi Burgiel Dept. of Mathematics, Bridgewater State University 24 Park Avenue, Bridgewater, MA 02325, USA [email protected] Abstract This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra. Introduction Many authors and educators have used origami, the art of paper folding, to provide concrete examples moti- vating mathematical problem solving. [3, 2] In unit origami, multiple sheets are folded and combined to form a whole; the Sonobe unit is a classic module in this art form. This workshop describes the construction and assembly of the star-building unit1, highlights a small selection of the many geometric questions motivated by this process, and introduces participants to the strictly convex deltahedra. Proficiency in geometry and origami is not required to enjoy this event. About the Unit Three Sonobe units interlock to form a pyramid with an equilateral triangular base as shown in Figure 1.
    [Show full text]
  • Introduction to Crystallography and Mineral Crystal Systems(4)
    Introduction to Crystallography and Mineral Crystal Systems by Mike and Darcy Howard Part 4: The Tetragonal System So you didn't get enough punishment in Article 3 and are back for more. Don't say we didn't warn you that thinking about all this crystallography stuff is addictive and will warp your sense of priorities! You either love these articles by now or are totally masochistic. The lack of geometrical reasoning necessary to understand crystallography and symmetry is what drove a lot of college wannabe geologists into the College of Business! Let's begin... Our discussion of the TETRAGONAL SYSTEM starts by examining the tetragonal axial cross and comparing it to the isometric axial cross (Article 3). Remember that in the isometric system all 3 axes were the same length and at right angles to each other. In the tetragonal system, we retain the same angular relationships, but vary the length of the vertical axis, allowing it to be either longer or shorter than the other two. We then relabel the vertical axis as c, retaining the same positive and negative orientation of this axis (see fig. 4.1a and 4.1b) As to the Hermann-Mauguin notation for the tetragonal system, the first part of the notation (4 or -4) refers to the c axis and the second or third parts refer to the a1 and a2 axes and diagonal symmetry elements, in that order. The tetragonal prism and pyramid forms have the symmetry notation 4/m2/m2/m. First, I want to consider the tetragonal prisms. There are 3 of these open forms consisting of the 1st order, 2nd order, and ditetragonal prisms.
    [Show full text]
  • The Most Chiral Disphenoid, Pp. 375-384
    MATCH MATCH Commun. Math. Comput. Chem. 73 (2015) 375-384 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 The Most Chiral Disphenoid∗ Michel Petitjean University Paris 7, INSERM UMR-S 973, MTi, 35 rue H´el`eneBrion, 75205 Paris Cedex 13, France [email protected], [email protected] (Received September 30, 2014) Abstract We calculate the analytical expression of the chiral index of the disphenoids and we show that this chiral index has a maximum value. It is reached when the disphenoids havep their four triangular faces congruent to a triangle with squared sidelengths ratios 1 : 3- 2=2 : 3. To our knowledge it is the first time that a maximal chirality three dimensional set is characterized in the non labeled case. 1 Introduction There were many attempts to quantify the geometric degree of chirality of a conformer or of a set of points since more than a century (see [1] for a review). Although it is clearly understood that the minimal degree of chirality corresponds to achirality, it is unclear what could be the maximal degree of chirality, as noticed by Fowler [2]. We point out that we do not look for maximizing physical quantities related to circular dichroism or optical rotatory power, these latter having no simple relation with a geometric degree of chirality, in some mathematical sense. Since several chirality measures exist, several maximally chiral sets are potentially expected. This is exemplified by the case of triangles (i.e. sets of three non labeled points in the plane), for which several maximally chiral triangles were proposed [3{8].
    [Show full text]
  • Exploring Noble Polyhedra with the Program Stella4d
    Bridges 2020 Conference Proceedings Exploring Noble Polyhedra With the Program Stella4D Ulrich Mikloweit Essen, Germany; [email protected] Abstract This paper shows how noble polyhedra can be produced and investigated with the software Stella4D. In a very easy way the results of Max Brückner can be reproduced and verified and many previously unpublished noble solids could be found. Introduction Polyhedra consisting of only one type of faces are called isohedral, and those consisting of only one type of vertices are called isogonal. An example for the first category is a triangular bipyramid, one for the latter category is the icosidodecahedron, see Figure 1. (a) (b) Figure 1: (a) Triangular Bipyramid, (b) Icosidodecahedron. If both properties occur in the same shape this is called a noble polyhedron. The most well-known noble polyhedra are the five Platonic solids. They are convex and they have the additional property that their faces are all regular. There is only one additional shape known which is both noble and convex, or more accurately: an infinite continuum of distorted shapes. These are the disphenoids. You can create a disphenoid by stretching a tetrahedron along a 2-fold axis. You get an irregular tetrahedron made of isosceles triangles where all faces and vertices are the same (Figure 2 (a)). As you can vary the degree of stretching, there are infinitely many of them. There are also nonconvex noble polyhedra, with intersecting faces. Among these are the four Kepler- Poinsot polyhedra the faces of which are regular too. One of them has equilateral triangles, one has regular pentagons and two of them are set up by regular pentagrams.
    [Show full text]
  • Polygons, Polyhedra, Patterns & Beyond
    POLYGONS, POLYHEDRA, PATTERNS & BEYOND Ethan D. Bloch Spring 2015 i ii Contents To the Student v Part I POLYGONS & POLYHEDRA 1 1 Geometry Basics 3 1.1 Euclid and Non-Euclid . 3 1.2 Lines and Angles ................................ 12 1.3 Distance .................................... 21 2 Polygons 25 2.1 Introduction................................... 25 2.2 Triangles .................................... 28 2.3 General Polygons ................................ 39 2.4 Area....................................... 49 2.5 The Pythagorean Theorem. 63 3 Polyhedra 73 3.1 Polyhedra – The Basics . 73 3.2 Regular Polyhedra. 78 3.3 Semi-Regular Polyhedra. 81 3.4 Other Categories of Polyhedra. 84 3.5 Enumeration in Polyhedra. 87 3.6 Curvature of Polyhedra . 99 iv Contents Part II SYMMETRY & PATTERNS 105 4 Isometries 107 4.1 Introduction................................... 107 4.2 Isometries – The Basics . 110 4.3 Recognizing Isometries . 119 4.4 Combining Isometries. 125 4.5 Glide Reflections . 130 4.6 Isometries—The Whole Story . 136 5 Symmetry of Planar Objects and Ornamental Patterns 147 5.1 Basic Ideas ................................... 147 5.2 Symmetry of Regular Polygons I . 154 5.3 Symmetry of Regular Polygons II . 160 5.4 Rosette Patterns . 170 5.5 Frieze Patterns.................................. 178 5.6 Wallpaper Patterns . 187 5.7 Three Dimensional Symmetry . 204 6 Groups 215 6.1 The basic idea.................................. 215 6.2 Clock arithmetic. 218 6.3 The Integers Mod n ...............................223 6.4 Groups .....................................228 6.5 Subgroups.................................... 237 6.6 Symmetry and Groups. 241 Suggestions for Further Reading 245 Bibliography 251 To the Student I have written these lecture notes because I have not found any existing text for Math 107 that was adequate in terms of the choice of material, level of difficulty, organization and exposition.
    [Show full text]